Existence and Global Attractivity of Positive Periodic Solutions for The Neutral Multidelay Logarithmic Population Model with Impulse

Definition 1 (see [21].)A function N : R → (0, +∞) is said to be a positive solution of (1) and (2), if the following conditions are satisfied:

• (a)

  N(t) is absolutely continuous on each (t_k, t_k+1);

• (b)

  for each $$ and $$ exist and $$;

• (c)

  N(t) satisfies the first equation of (1) and (2) for almost everywhere (for short a.e.) in [0, ∞]∖{t_k} and satisfies $$ for t = t_k,  k ∈ Z₊ = {1,2, …}.

Definition 2. The system (1) is said to be globally attractive, if there exists a positive solution x(t) of (1) such that lim _t→+∞ | x(t) − y(t)| = 0 for any other positive solution y(t) of (1).

Lemma 3. The region R = {N(t) : N(0) > 0,  t ≥ 0} is invariant with respect to (1).

Proof. In view of biological population, we obtain N(0) > 0. By the system (1), we have

Then, the solution of (1) and (2) is positive.

Under the above hypotheses (H₁)–(H₃), we consider the following neutral nonimpulsive system:

with the following initial conditions: where By a solution y(t) of (10) and (11), it means an absolutely continuous function y(t) defined on [−τ, 0] that satisfies (10) a.e., for t ≥ 0, and y(ξ) = φ(ξ),  y^′(ξ) = φ^′(ξ) on [−τ, 0].

The following lemmas will be used in the proofs of our results, and the proof of the lemma is similar to that of Theorem 1 in [13].

Lemma 4. Suppose that (H₁)–(H₄) hold. Then,

• (1)

  if y(t) is a solution of (10) and (11) on [−τ, +∞), then $$ is a solution of (1) and (2) on [−τ, +∞);

• (2)

  if N(t) is a solution of (1) and (2) on [−τ, +∞), then $$ is a solution of (10) and (11) on [−τ, +∞).

Proof.   (1) It is easy to see that $$ is absolutely continuous on every interval (t_k, t_k+1],  t ≠ t_k,  k = 1,2, …,

On the other hand, for any t = t_k,  k = 1,2, …, thus, It follows from (13)–(15) that N_i(t) is a solution of (1) and (2).

(2) Since $$ is absolutely continuous on every interval (t_k, t_k+1],  t ≠ t_k, k = 1,2, …, and in view of (15), it follows that for any k = 1,2, …,

which implies that y(t) is continuous on [−τ, +∞). It is easy to prove that y(t) is absolutely continuous on [−τ, +∞). Similar to the proof of (1), one can check that $$ are solutions of (10) and (11) on [−τ, +∞). The proof of Lemma 3 is completed.

Definition 5 (see [23].)Let U be a bounded subset in X. Define that

where diam (U_i) denotes the diameter of the set U_i, obviously, 0 ≤ α_X(U) < ∞. So, α_X(U) is called the (Kuratowski) measure of noncompactness of X.

Definition 6 (see [23].)Let X, Y be two Banach spaces and D ⊂ X; a continuous and bounded map T : D → Y is called k-set contractive if for any bounded set U ⊂ D one has

T is called strict-set-contractive if it is k-set contractive for some 0 ≤ k < 1.

For a Fredholm operator L : X → Y with index zero, according to [9, 24], we define

Lemma 7 (see [9], [24].)Let L : X → Y be a Fredholm operator with zero index and a ∈ Y be a fixed point. Suppose that N : Ω → Y is called a k-set contractive with k < l(L), where Ω ⊂ X is bounded, open, and symmetric about 0 ∈ Ω. Further, one also assumes that

• (1)

  Lx ≠ λNx + λa,  for x ∈ ∂Ω,  λ ∈ (0,1);

• (2)

  [QN(x) + Qa, x] [QN(−x) + Qa, x] < 0, for x ∈ Ker L⋂ ∂Ω;

where [·, ·] is a bilinear form on Y  ×  X and Q is the projection of Y onto Coker(L), where Coker(L) is the cokernel of the operator L. Then, there is an $$ such that Lx − Nx = a.

Lemma 8 (see [25].)The differential operator L is a Fredholm operator with index zero and satisfies l(L) ≥ 1.

Lemma 9. Let r₀,  r₁ be two positive constants and $$. If $$, then $$ is a k-set contractive map.

Lemma 10 (see [7], [9].)Suppose that $$ and τ^′(t) < 1, t ∈ [0, ω], then the function t − τ(t) has a unique inverse μ(t) satisfying μ ∈ C(R, R) with μ(a + ω) = μ(a) + ω for all a ∈ R, and if g ∈ PC_ω,   τ^′(t) < 1,   t ∈ [0, ω], then g(μ(t)) ∈ PC_ω.

Suppose that x(t) is a differently continuous ω-periodic function on R with (ω > 0), then to any $$.

Remark 12. From Lemma 9, we get that ζ_i(ω) = ζ_i(0) + ω,  ξ_l(ω) = ξ_l(0) + ω,  i = 1,2, …, n; l = 1,2, …, p; then,

Similarly, Thus,

Theorem 13. In addition to (H₁)–(H₃), suppose that the following conditions hold:

•  

  (H₄) there exists a constant η > 0 such that |Γ(t)| ≥ η, for all t ∈ [0, ω], where Γ(t) is defined by (23);

•  

  (H₅) if $$, and $$.

Then, (1) has at least one positive ω-periodic solution.

Proof. Let u(t) be an arbitrary ω-periodic solution of the operator equation as follows:

where L, N defined by (21) and (22), respectively. Then, u(t) satisfies the following operator equation: Integrating both sides of (28) over [0, ω], we have Let t − σ_i(t) = s, then t = ζ_i(s) and By Lemma 10, we have Similarly, we have Substituting (31) and (32) into (29), we have Considering assumption (H₄), we know that |Γ(t)| ≥ η > 0, and then, it follows from the integro mean value theorem that there exists a α ∈ [0, ω] satisfying By Lemma 11, we can get that Multiplying both sides of (28) by u^′(t) and integrating them over [0, ω], we have By using the Cauchy-Schwarz inequality, we get that Meanwhile, we see that Substituting (38) and (35) into (37), we can find that which gives that From (H₅), we get that $$. Then, there exists a constant N > 0 such that From (35) and the $$ inequality, we obtain that Again from (28), we get that From condition $$, it is easy to see that Now, we take $$ and $$. Then, $$. So by (42) and (44), we can find that all the conditions of Lemma 4 except (2) hold. In what follows, we will prove that condition (2) of Lemma 4 is also satisfied. In order to do this, we defined a bounded bilinear form on $$ by [·, ·] as the following $$. Also, we defined Q : y → Coker(L) as $$. It is obvious that Without loss of generality, suppose that u ≡ N₃; then, since $$, then By (46), we get that Therefore, by using Lemma 4, we obtain that (1) has at least one positive ω-periodic solution; the proof of Theorem 13 is completed.

Corollary 14. Suppose that (H₁)–(H₃) and the following conditions hold:

•  

  $$ there exists a constant η > 0 such that |Γ(t)| ≥ η, for all t ∈ [0, ω], where Γ(t) is defined by (23);

•  

  $$ if $$.

Then (1) has at least one positive ω-periodic solution.

Corollary 15. Suppose that (H₁)–(H₃) and the following conditions hold:

•  

  $$ there exists a constant η > 0 such that |Γ(t)| ≥ η, for all t ∈ [0, ω], where Γ(t) is defined by (23);

•  

  $$ if $$.

Then, (1) has at least one positive ω-periodic solution.

Theorem 16. Suppose that (H₁)–(H₅) and the following conditions hold:

•  

  (H₆) there is a positive constant μ such that

  $$ ()

•  

  (H₇) $$, as t → +∞.

Then, the positive ω-periodic solution of (1) is globally attractive, where $$.

Proof. Suppose that y(t) = e^v(t) is a positive ω-periodic solution of (10), $$ is another positive solution of (10). Similar to (17), we have

Let v(t) − v^*(t) = w(t); then, Multiply both sides of (51) with $$ and then integrate from 0 to t to obtain that Then, Meanwhile, we see that where Substituting (55) into (54), we get that Therefore, we have From (H₆), we have From (H₇), we have thus, v(t) → v^*(t), as t → +∞; that is, the positive ω-periodic solution of (10) is globally attractive; by Definition 2, the positive ω-periodic solution of (1) is globally attractive. The proof is completed.

Corollary 17. Suppose that the following conditions hold:

•  

  $$ r(t),  a(t),  b_i(t),  c_j(t),  d_l(t),  σ_i(t), τ_l(t)  (i = 1,2, …, n; j = 1,2, …, m; l = 1,2, …, p) are all positive ω-periodic continuous functions with σ_i(t) ≥ 0,  τ_l(t) ≥ 0, t ∈ [0, ω], $$, $$, for all i = {1,2, …, n}, for all l = {1,2, …, p}; furthermore, d_l(t) ∈ C¹(R, R), $$, for all j = {1,2, …, m}, for all l = {1,2, …, p};

•  

  $$ there exists a constant η > 0 such that |Γ^*(t)| ≥ η, for all t ∈ [0, ω], where Γ^*(t) is defined by the following:

  $$ ()

•  

  $$ if $$, and $$.

Then, (61) has at least one positive ω-periodic solution.

Corollary 18. Suppose that $$ and the following conditions hold:

•  

  $$ there is a positive constant μ such that

  $$ ()

•  

  $$ $$, as t → +∞.

Then, the positive ω-periodic solution of (61) is globally attractive, where $$.

Remark 19. One could easily see that the systems (5)–(7) are all special cases of the system (61); we can get the similar results, and we omit them here. Hence, our results improve and generalize the corresponding results in [4–7].

Corollary 20. Suppose that (H₁)–(H₃) and the following conditions hold:

•  

  (H₄) there exists a constant η > 0 such that |Γ(t)| ≥ η, for all t ∈ [0, ω], where Γ(t) is defined by the following:

  $$ ()

•  

  (H₅) if $$, and |1 − τ^′|₀ | D|₀ < 1.

Then, (64) has at least one positive ω-periodic solution, where

Corollary 21. Suppose that (H₁)–(H₅) and the following conditions hold:

•  

  (H₆) there is a positive constant μ such that

  $$ ()

•  

  (H₇) $$, as t → +∞.

Then, the positive ω-periodic solution of (64) is globally attractive, where H(t): = [D^′(t) + D(t)A(t)] [1 − τ^′(t)] + D(t)τ^′′(t)/[1 − τ^′(t)] ².

Remark 22. The results in the work show that by means of appropriate impulsive perturbations, we can control the dynamics of these equations.